Subsurface geological modeling involves estimating parameters of interest in a 3-d model which is used for development planning and production forecasting.
A geological reservoir model should, as far as possible, be in conformity with the geological rules or features that are specific to the depositional environment of the reservoir of interest. The geomodel also, as far as possible, needs to be conditioned to all quantitative information from well cores and logs, seismic attributes, well tests and production history, etc. Thirdly, because of our incomplete knowledge about the subsurface reservoir, there is always a degree of uncertainty that needs to be accounted for in the reservoir modeling process.
In this context, several geostatistical or probabilistic methods have been developed for geological reservoir modeling. These methods allow the building of models that are representative of various depositional environments, with various types of data being integrated into the models. In addition, the nature of the probabilistic approach makes it possible to account for uncertainty.
Once a geostatistical reservoir model is chosen to describe the reservoir of interest, one can generate, potentially, an infinite number of so-called realizations. Each realization consists of a matrix comprising several values (representing parameters such as porosity and permeability) associated with each of a large number of cells distributed over the volume of the reservoir. Only for a relatively small number of these cells will the values be known with relative certainty (namely those cells which contain parameters which have been measured). For the remaining cells, the values are estimates based on the geostatistical modeling process. Each realization will have a different set of estimated values, each realization having been generated using a different, but equally valid, random seed.
The model is used to make predictions of e.g. flow rates and pressures in wells; these are sometimes known as the “dynamic responses” predicted by the model. Data (e.g. flow rate data) will be gathered from the wells over time once the reservoir is in production and, generally, these will differ from the predicted values generated by the model. History matching is a process by which new realizations of the model are generated which predict the correct current values for e.g. flow rate and pressure, and can therefore be assumed to be more accurate and to make more accurate predictions of these values for the future. In any history matching process creating updated realizations, a good goal is to preserve the geological features and statistical data on which the model is based.
In history matching, an initial realization is modified so that the simulated dynamic responses which it predicts match with the measured ones. As mentioned above, for any modification, it is helpful if the statistical data and geological features inherent in the model are preserved. A practical reservoir model will have in excess of a million cells each associated with several values; arbitrarily changing the values in such a model is both unfeasible because there are simply too many options and also will not ensure that the geological and statistical integrity of the model is preserved. What would be helpful would be a mathematical tool which allows easy adjustments to be made which vary the values in such a way that the geological features and statistics are preserved.
U.S. Pat. No. 6,813,565 (I.F.P.) discloses a history matching technique. This technique is sometimes known as the gradual transformation method. A number of uniform random fields Ui, are generated from respective random seeds Si. Each random field is used to generate a corresponding realization Ri. For history matching, a given realization is modified by modifying the uniform field which gave rise to it:Ui+ΔU→Ri+ΔR 
The '565 disclosure describes linearly combining a number of fields with combination coefficients ri. In the described history matching process, the combination coefficients are adjusted to vary the properties of the resulting modified uniform field and hence the realization resulting from that field. '565 describes a complex algorithm which involves converting each uniform field into a Gaussian field using a Gaussian score transformation function (sometimes called a normal score transformation or a Gaussian anamorphosis) prior to performing the linear combination, and then using an inverse Gaussian score transformation to derive a modified uniform field. This is illustrated below:                Si→Ui (Create a number of uniform fields from seeds Si)        G(Ui)→Yi (Gaussian score transformation on each uniform field)        ΣriYi→Ymod (Combine Gaussian fields-produce modified field)        G−1(Ymod)→Umod (Inv. Gaussian score transformation to get uniform field)        
In this approach, the parameters ri are manipulated to produce a realization which produces simulated dynamic responses matching those of the reservoir itself.
A problem arises because a modified random number field resulting from a linear combination of uniform fields (with combination coefficients ri) is not itself a uniform field. In contrast, when Gaussian fields are combined linearly, it results in another Gaussian field being produced. For this reason, the algorithm described in '565 is relatively complex, involving a Gaussian score transformation of each uniform field to be combined, followed by inverse Gaussian score transformation of the modified Gaussian field produced by the linear combination step.
An alternative approach to the gradual deformation method is described in Caers, J. Geostatistical history matching under training-image based geological model constraints. Society of Petroleum Engineers Annual Technical Conference and Exhibition, 29 Sep.-2 Oct. 2002. In this paper, an iterative approach is described in which one parameter rD is used to make successive realizations each approaching closer to a best match with the measured reservoir data. Each iteration allows for the combination of only two realizations, which means the space of possible solutions is relatively small. This technique is sometimes known as the probability perturbation method (P.P.M.) and was initially proposed by Jef Caers of Standard University.
In contrast to the gradual deformation method,                The P.P.M. is limited to geomodel realizations generated by the sequential simulation algorithm;        It allows the combining of only two realizations at a time; and        There is no mathematical proof that the combined realization preserves the spatial statistics of the predefined geomodel.        